In my opinion the best way to convert of shines vs. Falco or Fox is to wave-shine in to Thunder's Combo
Thunder's can be countered with SDI'ing the jab up and following with Shine or aerial- but is worth learning if your opponent doesn't counter it and you don't like tech-chasing.
Tech-chasing is something you can improve on in multiple ways:
Some people react tech-chase, which imo is the hardest method- if your timing isn't perfect your opponent can side-step/jab or shine the approach.
You can also tech-chase through reads if you notice your opponent has patterns.
I like 'semi-reaction' tech chasing, in which I'll prepare for a grab if they choose a certain option, and cover the others with a shffl'd aerial into shine, (eg. Drill shine), so that it's a little safer than risking a whiffed or badly timed grab.
You can tech-chase sufficiently regardless of your method so long as it qualifies as a good punish.
Just keep in mind that a bad tech-chase can sometimes be worse than not tech-chasing at all.